THE TATE-VOLOCH CONJECTURE FOR DRINFELD MODULES
DRAGOS GHIOCA
Abstract. We study the v-adic distance from the torsion of a Drinfeld module to an affine
variety.
1. Introduction
For a semi-abelian variety S and an algebraic subvariety X ⊂ S, the Manin-Mumford
conjecture characterizes the subset of torsion points of S contained in X. The Tate-Voloch
conjecture characterizes the distance from X of a torsion point of S not contained in X.
Let Cp be the completion of a fixed algebraic closure Qalg
p of Qp . Let λ(·, X) be the p-adic
proximity to X function as defined in [11] (see also our definition of v-adic distance to an
affine subvariety). Tate and Voloch conjectured:
Conjecture 1.1 (Tate,Voloch). Let G be a semi-abelian variety over Cp . Let X ⊂ G be a
subvariety defined over Cp . Then there is a constant N ∈ N such that for any torsion point
ζ ∈ G(Cp ) either ζ ∈ X or λp (ζ, X) ≤ N .
The above conjecture was proved by Thomas Scanlon for all semi-abelian varieties defined
over Qalg
p (see [11] and [12]).
In this paper we prove two Tate-Voloch type theorems for Drinfeld modules. Our motivation is to show that yet another question for semi-abelian varieties has a counterpart for
Drinfeld modules (see [13] and [5] for the Manin-Mumford theorem for Drinfeld modules of
generic characteristic and see [4] for the Mordell-Lang theorem for all Drinfeld modules).
In Section 2 we state our results. Our first result (Theorem 2.7) shows that if a torsion
point of a Drinfeld module φ : A → K{τ } is close w-adically to a variety X with respect to
all places w extending a fixed place v of the ground field K, then the torsion point lies on
X. We prove Theorem 2.7 in Section 3. Our bound for how “close w-adically to X” means
“lying on X” is effective. Our second result (Theorem 2.10) refers to proximity with respect
to one fixed extension of a place v of K. We will prove Theorem 2.10 in Section 4. We also
note that due to the fact that in Theorem 2.10 we work with a fixed extension of a place
of K, there is a different normalization for the valuation we are working as opposed to the
setting in Theorem 2.7.
I thank Thomas Scanlon for a conversation regarding this paper. I thank Damian Roessler
for bringing to my attention the Tate-Voloch conjecture. I thank the anonymous referee for
his comments.
2. Statement of our main results
Before stating our results we introduce the definition of a Drinfeld module (for more
details, see [3]).
1
2000 AMS Subject Classification: Primary, 11G09; Secondary, 11G50
1
Let p be a prime number and let q be a power of p. We let C be a nonsingular projective
curve defined over Fq and we fix a closed point ∞ on C. Then we define A as the ring of
functions on C that are regular everywhere except possibly at ∞.
We let K be a field extension of Fq and we fix an algebraic closure of K, denoted K alg .
We fix a morphism i : A → K. We define the operator τ as the power of the usual
Frobenius with the property that for every x ∈ K alg , τ (x) = xq . Then we let K{τ } be
the ring of polynomials in τ with coefficients in K (the addition is the usual one, while the
multiplication is the composition of functions).
A Drinfeld module over K is a ring morphism φ : A → K{τ } for which the coefficient of
τ 0 in φa is i(a) for every a ∈ A, and there exists a ∈ A such that φa 6= i(a)τ 0 . We call φ a
Drinfeld module of generic characteristic if ker(i) = {0} and we call φ a Drinfeld module of
finite characteristic if ker(i) 6= {0}. In the generic characteristic case we assume i extends
to an embedding of Frac(A) (which is the function field of the projective nonsingular curve
C) into K. In the finite characteristic case, we call ker(i) the characteristic ideal of φ.
alg
For every nonzero a ∈ A, let the a-torsion φ[a]
S of φ be the set of all x ∈ K such that
φa (x) = 0. Let the torsion submodule of φ be a∈A\{0} φ[a].
For every g ≥ 1, let φ act diagonally on Gga . An element (x1 , . . . , xg ) ∈ (K alg )g is called a
torsion element of φ, if for every i ∈ {1, . . . , g}, xi ∈ φtor .
For each field extension L of K and for each valuation w on L we define the w-adic distance
to an affine subvariety X ⊂ Gga defined over L.
Definition 2.1. Let IX be the vanishing ideal in L[X1 , . . . , Xg ] of X. Let Rw ⊂ L be the
valuation ring of w. If P ∈ Gga (L), then the w-adic distance from P to X is
(1)
λw (P, X) := min{w(f (P )) | f ∈ IX ∩ Rw [X1 , . . . , Xg ]}.
We denote by MK the set of all discrete valuations on K. Similarly, for each field extension
L of K we also denote by ML the set of all discrete valuations on L. Finally, we note that
unless otherwise stated, each valuation is normalized so that its range is precisely Z ∪ {+∞}
(our convention is that the valuation of 0 is +∞). Our Theorem 2.7 is valid for all fields K
equipped with a coherent good set of valuations.
Definition 2.2. We call a subset U ⊂ MK equipped with a function d : U → R>0 a good
set of valuations if the following properties are satisfied
(i) for each nonzero x ∈ K, there are finitely many v ∈ U such that v(x) 6= 0.
(ii) for each nonzero x ∈ K,
X
d(v) · v(x) = 0.
v∈U
The positive real number d(v) will be called the degree of the valuation v. When we say
that the positive real number d(v) is associated to the valuation v, we understand that the
degree of v is d(v).
When U is a good set of valuations, we will refer to property (ii) as the sum formula for
U.
2
Definition 2.3. Let v ∈ MK of degree d(v). We say that the valuation v is coherent if for
every finite extension L of K,
X
(2)
e(w|v)f (w|v) = [L : K],
w∈ML
w|v
where e(w|v) is the ramification index and f (w|v) is the relative degree between the residue
field of w and the residue field of v.
Condition (2) says that v is defectless in L. In this case, we also let the degree of any
w ∈ ML , w|v be
f (w|v)d(v)
.
(3)
d(w) =
[L : K]
Definition 2.4. We let UK be a good set of valuations on K. We call UK a coherent good
set of valuations if for every v ∈ UK , the valuation v is coherent.
Remark 2.5. Using the argument from page 9 of [10], we conclude that in Definition 2.4,
if for each finite extension L of K we let UL ⊂ ML be the set of valuations lying above
valuations in UK , then UL is a good set of valuations.
Example 2.6. Let V be a projective, regular in codimension 1 variety defined over a finite
field. Then the function field F of V is equipped with a coherent good set of valuations
associated to each irreducible divisor of V . Hence every finitely generated field is equipped
with at least one coherent good set of valuations (different sets of valuations correspond to
different projective, regular in codimension 1 varieties with the same function field). For
more details see [10] or Chapter 4 of [3].
We prove the following Tate-Voloch type theorem for Drinfeld modules.
Theorem 2.7. Assume UK is a coherent good set of valuations on K and let v ∈ UK have
degree d(v). Let φ : A → K{τ } be a Drinfeld module. Let X ⊂ Gga be a closed K-subvariety
of the g-dimensional affine space.
There exists a constant C > 0 (depending on φ, X and d(v)) such that for every finite
extension L of K and for every torsion point P ∈ Gga (L) of φ, either P ∈ X(L) or there
exists w ∈ ML lying over v such that λw (P, X) ≤ C · e(w|v).
Remark 2.8. There are two significant differences between our Tate-Voloch type theorem and
Conjecture 1.1. We show that a torsion point of the Drinfeld module is on X if it is close to
X with respect to all extensions of a fixed valuation v of K, not only with respect to one fixed
extension of v. We will show in Example 2.9 that we cannot always expect proximity of P
to X with respect to one fixed extension of v imply that P lies on X. The second difference
between our Theorem 2.7 and Conjecture 1.1 is purely technical. Because we normalized all
valuations so that their ranges equal Z, we need to multiply by the corresponding ramification
index the constant C in Theorem 2.7.
Example 2.9. Let φ be any Drinfeld module of generic characteristic and let v∞ be a
valuation on K extending the valuation on Frac(A) associated to the closed point ∞ ∈ C.
alg
We let K∞ be a completion of K with respect to v∞ . Then φtor ⊂ K∞
is not discrete
with respect to v∞ (see Section 4.13 of [7]). Hence there exist nonzero torsion points of φ
arbitrarily close to X := {0} in the v∞ -adic topology.
3
For the remainder of Section 2 we fix a valuation v on K (we do not require anymore
that v belongs to a good set of valuations on K nor that v is coherent). We let Kv be the
completion of K at v. We fix an algebraic closure Kvalg of Kv and extend v to a valuation of
Kvalg . In this case, the value group of v is Q. We define as in (1) the v-adic distance from a
point P ∈ Gga (Kvalg ) to a fixed affine variety X defined over Kvalg .
Our Theorem 2.10 characterizes the distance from φgtor to a fixed point of Gga (Kvalg ). Our
theorem is an analogue for Drinfeld modules of a theorem of Mattuck (see [8]).
Theorem 2.10. Let φ : A → K{τ } be a Drinfeld module. Let v be a place of K. If φ is a
Drinfeld module of generic characteristic, then assume v does not lie over the valuation v∞
of Frac(A), which is associated to the closed point ∞ ∈ C. Let g ≥ 1.
Then for every Q ∈ Gga (Kvalg ) there exists a positive constant C depending on φ, v and Q
such that for each P ∈ φgtor either P = Q or λv (P, Q) < C.
Note that as shown in Example 2.9, if φ has generic characteristic, then Theorem 2.10
does not hold if v extends the place v∞ of Frac(A). If φ has finite characteristic, there is no
restriction on v in Theorem 2.10.
3. Proximity with respect to all extensions of a valuation v
We work under the assumption that there exists a coherent good set of valuations UK on
K. We first construct the set of local heights associated to the places in UK and then we
define the global height. All our valuations in this section are normalized so that their value
group is Z.
For each finite extension field L of K and for each place w of L lying above a place in UK ,
we let w̃ : L → Z≤0 be defined as follows
w̃ := min{w, 0}.
Then the local height at w of any element x ∈ L is hw (x) := −d(w)w̃(x). We define the
global height of x as
X
hw (x).
h(x) :=
w
The above sum is a finite sum because there are finitely many w such that w(x) < 0 (see
condition (i) of Definition 2.2). Because UK is a coherent good set of valuations, the definition
of the global height of an element x does not depend on the particular choice of the field L
containing x (see for example Chapter 4 of [3]). The following two standard properties of
the height will be used in our proof.
Proposition 3.1. For each x, y ∈ K alg , the following are true:
(i) h(xy) ≤ h(x) + h(y).
(ii) h(x + y) ≤ h(x) + h(y).
Proof. The proof is immediate using the definition of height and the triangle inequality for
each valuation.
For a point P := (x1 , . . . , xg ) ∈ Gga (L), we define the local height of P at a place w of L
lying above a place in UK , as follows:
hw (P ) := max{hw (x1 ), . . . , hw (xg )}.
4
P
Then the global height of P is h(P ) := w hw (P ).
Next we define the heights associated to a Drinfeld module φ : A → K{τ } (see [3] for
more details). We fix a non-constant a ∈ A. For each finite extension L as above and for
each place w of L as above, we define
w̃(φan (x))
,
n→∞ deg(φan )
Vw (x) := lim
for each x ∈ L.
Then the canonical local height of x at w with respect to φ is b
h (x) := −d(w)Vw (x).
Pw b
b
Finally, the canonical global height of x with respect to φ is h(x) := w hw (x). By the same
reasoning as in [1] (see part 3) of Théorème 1) or in [9] (see part (2) of Proposition 1) we
can show that there exists a positive constant C0 such that for every x ∈ K alg ,
| h(x) − b
h(x)| ≤ C0 .
(4)
Moreover, the constant C0 is easily computable in terms of φ (see [9]).
For each point P := (x1 , . . . , xg ) ∈ Gga (L) and for each place w of L as above, we define
the canonical local height of P at w as b
hw (P ) := max{b
hw (x1 ), . . . , b
hw (xg )}. The canonical
P
hw (P ).
global height of P is b
h(P ) := w b
Using (4) and Proposition 3.1 we prove the following result.
Lemma 3.2. Let L be a finite extension of K and let f ∈ L[X1 , . . . , Xg ]. There exists a
constant C(f ) > 0 such that for every P ∈ Gga (K alg ), if P is a torsion point for φ, then
h(f (P )) ≤ C(f ).
Proof. Using Proposition 3.1 (ii), it suffices to prove Lemma 3.2 under the assumption that
α
f is a monomial. Hence, assume f := cX1α1 · · · · · Xg g for some c ∈ L and α1 , . . . , αg ∈ Z≥0 .
Let P = (x1 , . . . , xg ). We know that for each i, xi ∈ φtor . Hence b
h(xi ) = 0 for each i. Using
(4) we conclude that h(xi ) ≤ C0 for each i. Therefore, an application of Proposition 3.1 (i)
concludes the proof of our Lemma 3.2.
We proceed to the proof of Theorem 2.7.
Proof of Theorem 2.7. Let f1 , . . . , fm be a set of polynomials in K[X1 , . . . , Xg ] with integral coefficients at v, which generate the vanishing ideal of X. It suffices to prove that for
each such polynomial fi and for every finite extension L of K and for every torsion point
i)
P ∈ Gga (L), either fi (P ) = 0 or there exists a place w|v of L such that w(fi (P )) ≤ C(f
e(w|v),
d(v)
where C(fi ) is the constant corresponding to fi as in Lemma 3.2. Then we obtain Theoi)
.
rem 2.7 with C := maxi C(f
d(v)
Assume for some i ∈ {1, . . . , m} and for some torsion point P ∈ Gga (L), w(fi (P )) >
C(fi )
e(w|v) for every place w|v of L. Then
d(v)
(5)
X
d(w) · w(fi (P )) >
w|v
C(fi ) X d(v)f (w|v)e(w|v)
C(fi ) X
d(w)e(w|v) =
= C(fi ) > 0
d(v)
d(v)
[L : K]
w|v
w|v
P
because w|v f (w|v)e(w|v) = [L : K], as v is a coherent valuation. If fi (P ) 6= 0, then (5)
yields that the set S of places of L lying above places in UK for which fi (P ) is non-integral,
5
is non-empty. Moreover, using (5) and the sum formula for the nonzero element fi (P ) ∈ L,
we conclude
X
(6)
d(w) · w(fi (P )) < −C(fi ).
w∈S
Therefore, by the definition of the local heights we get
X
(7)
hw (fi (P )) > C(fi ).
w∈S
Using the definition of the global height and (7) we conclude h(fi (P )) > C(fi ). This last
inequality contradicts Lemma 3.2 because P is a torsion point. This shows that fi (P ) =
0 assuming fi (P ) is close w-adically to 0 for each w|v. This concludes the proof of our
Theorem 2.7.
Remark 3.3. Theorem 2.7 cannot be strenghtened to ask that proximity of P to X with
respect to one extension w of v would guarantee that P ∈ X (see Example 2.9). However,
even if we want to strenghten Theorem 2.7 by assuming proximity with respect to only one
extension of v (under the extra assumption that v does not lie over v∞ ), our proof would not
extend. We use in a crucial way in (5) that P is close to X with respect to all extensions of
v. If we would know this information about only one place w, this would not guarantee that
fi (P ) has “sufficiently many zeros” (as described in (5)). In turn, this would not yield that
fi (P ) has “sufficiently many poles” (as in (6)) and hence, we would not obtain a contradiction
regarding the height of fi (P ) (as in (7)). We believe that the question of proximity with
respect to one extension of a valuation v (which does not extend v∞ ) is a difficult question
and we also believe answering this question would involve new methods.
Remark 3.4. Because the constants C(fi ) from the proof of Theorem 2.7 are easily computable in terms of the polynomials fi and in terms of the constant from (4), then the
constant C from the conclusion of Theorem 2.7 is effective.
4. Proximity with respect to a fixed extension of the valuation v
In this Section 4 we work under the hypothesis that the valuation v of K does not extend
the valuation v∞ of Frac(A) in case φ : A → K{τ } is a Drinfeld module of generic characteristic. We also work with a fixed completion Kv of K at v and with its algebraic closure
Kvalg . In this section, the value group of our valuation v is Q, while its restriction to K has
value group Z.
We first reduce Theorem 2.10 to the following Lemma 4.1.
Lemma 4.1. Let φ : A → K{τ } be a Drinfeld module and let v be a discrete valuation on
K. If φ has generic characteristic, assume moreover that v does not lie over the place v∞ of
Frac(A). Then there exists a positive constant Cv depending only on φ and v such that in
the ball
{x ∈ Kvalg | v(x) ≥ Cv }
there are no nonzero torsion points of φ.
Lemma 4.1 shows that for each place v which does not lie over v∞ (if φ has generic
characteristic), φtor is discrete in the v-adic topology. If φ has finite characteristic, then φtor
is discrete with respect to each valuation v (without any restriction). Moreover, as it will be
shown in the proof of Lemma 4.1, the constant Cv is easily computable in terms of φ and v.
6
Proof of Theorem 2.10. We prove Theorem 2.10 using the result of Lemma 4.1. Let Q :=
(y1 , . . . , yg ) and let L := Kv (Q). Let βi := max{0, −v(yi )} for each i ∈ {1, . . . , g}. For each
i, let γi ∈ L be an element of valuation equal to βi . Then for each i ∈ {1, . . . , g}, the linear
polynomial γi (Xi − yi ) ∈ L[X1 , . . . , Xg ] has integral coefficients at v and vanishes at Q.
We know (see Lemma 5.2.5 of [3] or Lemma 4.12 of [6]) that there exists an absolute
constant Mv ≤ 0 depending only on φ and v such that for every torsion point x ∈ φtor ,
v(x) ≥ Mv (because otherwise, x has positive local height at v, contradicting the fact that
each local height of a torsion point is 0). Then for each point P := (x1 , . . . , xg ) ∈ φgtor , if for
some i ∈ {1, . . . , g},
v(yi ) = −βi < Mv ≤ v(xi ),
then v(xi − yi ) = v(yi ). In this case, λv (P, Q) ≤ v (γi (xi − yi )) = 0. Therefore, in case
for some i ∈ {1, . . . , g}, v(yi ) < Mv , we obtained an absolute upper bound for the v-adic
distance of a torsion point to Q.
Assume from now on in this proof that for every i ∈ {1, . . . , g}, v(yi ) ≥ Mv . Hence
βi ≤ −Mv . We compute the v-adic distance between a torsion point P := (x1 , . . . , xg ) ∈ φgtor
and Q. We obtain:
g
(8)
g
g
λv (P, Q) ≤ min v(γi (xi − yi )) = min (βi + v(yi − xi )) ≤ −Mv + min v(xi − yi ).
i=1
i=1
i=1
Therefore, in order to prove Theorem 2.10 it suffices to show that
g
min v(xi − yi )
i=1
is uniformly bounded from above when (x1 , . . . , xg ) ∈ φgtor \ {(y1 , . . . , yg )}. But Lemma 4.1
shows that for each i, there is at most one torsion point of φ in the ball
(9)
{x ∈ Kvalg | v(x − yi ) ≥ Cv },
because otherwise there would be at least one nonzero torsion point of φ in {x ∈ Kvalg |
v(x) ≥ Cv } after translating the ball in (9) by a torsion point of φ which lies inside the ball
from (9). Therefore, λv (P, Q) is indeed uniformly bounded from above for P ∈ φgtor \ {Q}
because there is at most one torsion point P ∈ φgtor such that λv (P, Q) > −Mv + Cv .
Remark 4.2. As discussed in Remark 3.3, the problem of proximity to an arbitrary variety
X of a torsion point with respect to a single extension of a valuation v seems to be a difficult
question. We note that the methods involved in our proof of Theorem 2.10 do not easily
generalize to the case of higher dimensional varieties X because then the vanishing ideal of
X would not be necessarily generated by linear polynomials. This would prevent us to have
a good control (as we had in (8)) on computing the v-adic distance to X of a torsion point.
We proceed to the proof of Lemma 4.1.
Proof of Lemma 4.1. We first choose t ∈ A satisfying certain properties according to the two
cases we have: φ has generic characteristic or not.
Case (i). φ has generic characteristic.
Let p be the nonzero prime ideal of A which is contained in the maximal ideal of the
valuation ring of v (we are using the fact that v does not lie over v∞ to derive that all the
elements of A are integral at v). We fix t ∈ p \ {0}.
Case (ii). φ has finite characteristic.
7
Let p be the characteristic ideal of φ. By the hypothesis for our Case (ii), p is nonzero.
We fix t ∈ p \P{0}.
r
i
Let φt =
i=r0 ai τ , where ar0 6= 0. In finite characteristic, r0 ≥ 1, while in generic
characteristic, r0 = 0 and v(a0 ) ≥ 1 (by our choice of t). We let Cv be the smallest positive
integer larger than all of the numbers from the following set:
v(ar )
v(ar0 ) − v(ai )
S := {− r0 0 } ∪ {
| r0 < i ≤ r}.
q −1
q i − q r0
We note that if φ has generic characteristic, then r0 = 0 and so, q r0 = 1. Then the denominator of the first fraction contained in S is 0. So, because the numerator −v(a0 ) ≤ −1, that
fraction equals −∞ and so, any integer is larger than it, i.e. if φ has generic characteristic,
we may disregard the first fraction in the definition of S. As we will see in our proof, that
first fraction will only be used in the finite characteristic case.
r Claim 4.3. If x ∈ Kvalg \ {0} satisfies v(x) ≥ Cv , then v(φt (x)) = v ar0 xq 0 > v(x) ≥ Cv .
In particular, φt (x) 6= 0.
v(a
)−v(a )
Proof of Claim 4.3. Because v(x) ≥ Cv , then for each i ∈ {r0 + 1, . . . , r}, v(x) > qri0−qr0 i .
Hence
v(ai ) + q i v(x) > v(ar0 ) + q r0 v(x) and so,
i
r (10)
v ai xq > v ar0 xq 0 for each i > r0 .
r Inequality (10) shows that v (φt (x)) = v ar0 xq 0 . In particular, this shows φt (x) does not
equal 0, because its valuation is not +∞ (both x and ar0 are nonzero numbers). Hence
(11)
v(φt (x)) = v(ar0 ) + q r0 v(x).
If φ has generic characteristic, then (11) shows that v(φt (x)) = v(a0 ) + v(x) ≥ 1 + v(x) > Cv .
If φ has finite characteristic, then using that
v(ar )
v(x) ≥ Cv > − r0 0 ,
q −1
r0
we conclude v(φt (x)) = v(ar0 ) + q v(x) > v(x) ≥ Cv .
Claim 4.3 shows that for every nonzero x ∈ Kvalg satisfying v(x) ≥ Cv , the sequence
{v(φtn (x))}n≥0 is strictly increasing. Hence, x ∈
/ φtor , because if x were torsion, then the
sequence {φtn (x)}n≥0 would contain only finitely many distinct elements. This concludes the
proof of Lemma 4.1.
References
[1] L. Denis, Canonical heights and Drinfeld modules. (French) Math. Ann. 294 (1992), no. 2, 213-223.
[2] L. Denis, The Lehmer problem in finite characteristic. (French) Compositio Math. 98 (1995), no. 2,
167-175.
[3] D. Ghioca, The arithmetic of Drinfeld modules. PhD thesis, UC Berkeley, May 2005.
[4] D. Ghioca, The Mordell-Lang theorem for Drinfeld modules. Internat. Math. Res. Notices 53 (2005),
3273-3307.
[5] D.Ghioca, Equidistribution for torsion points of a Drinfeld module. to appear in Math. Ann. (2006).
[6] D. Ghioca, The local Lehmer inequality for Drinfeld modules. submitted for publication (2005).
[7] D. Goss, Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete
(3) [Results in Mathematics and Related Areas (3)], 35. Springer-Verlag, Berlin, 1996.
8
[8] A. Mattuck, Abelian varieties over p-adic ground field. Ann. of Math. (2) 62 (1955), 92-119.
[9] B. Poonen, Local height functions and the Mordell-Weil theorem for Drinfeld modules. Compositio Mathematica 97 (1995), 349-368.
[10] J.-P. Serre, Lectures on the Mordell-Weil theorem. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. Aspects of Mathematics, E15. Friedr. Vieweg & Sohn,
Braunschweig, 1989. x+218 pp.
[11] T. Scanlon, p-adic distance from torsion points to semi-abelian varieties. J. Reine Angew. Math. 499
(1998), 225-236.
[12] T. Scanlon, The conjecture of Tate and Voloch on p-adic proximity to torsion. Internat. Math. Res.
Notices 17 (1999), 909-914.
[13] T. Scanlon, Diophantine geometry of the torsion of a Drinfeld module. J. Number Theory 97 (2002),
no. 1, 10-25.
Dragos Ghioca, Department of Mathematics, McMaster University, Hamilton Hall, Room
218, Hamilton, Ontario L8S 4K1, Canada
dghioca@math.mcmaster.ca
9